Soft Computing

, Volume 23, Issue 22, pp 11389–11398 | Cite as

Extensions of a tight function and their continuity in quantum logic

  • Mona Khare
  • Pratibha PandeyEmail author


In the present paper, a new variation of a nonnegative real-valued function \(\rho \) defined on a subfamily of a quantum logic P is proposed, and the notion of tightness of \(\rho \) is studied. Various crucial results are proved, and subsequently we have obtained extensions, viz. f-extension, \(\delta \)-extension and \(\sigma \)-extension of a tight function \(\rho ;\) continuity of extensions of \(\rho \) with respect to an approximating family in P is discussed.


Orthomodular lattices Continuity Tight functions Approximating families 



The authors are grateful to the anonymous referees for their valuable suggestions toward the improvement of the paper.


The second author acknowledges with gratitude the financial support by Department of Science and Technology (DST), New Delhi, India, under INSPIRE fellowship No. IF160721.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed consent

Informed consent was obtained from all individual participants (co-author) included in the study.


  1. Adamski W (1984) Extensions of tight set functions with applications in topological measure theory. Trans Am Math Soc 283:353–368MathSciNetCrossRefGoogle Scholar
  2. Adamski W (1987) On regular extensions of contents and measures. J Math Anal Appl 127:211–225MathSciNetCrossRefGoogle Scholar
  3. Avallone A, De Simone A (2001) Extensions of modular functions on orthomodular lattices. Ital J Pure Appl Math Soc 9:109–122MathSciNetzbMATHGoogle Scholar
  4. Beaver OR, Cook TA (1977) States on quantum logics and their connection with a theorem of Alexandroff. Proc Am Math Soc 67:133–134MathSciNetCrossRefGoogle Scholar
  5. Beltrametti EG, Cassinelli G (1981) The logic of quantum mechanics. Addison-Wesley, ReadingzbMATHGoogle Scholar
  6. Beran L (1984) Orthomodular lattices, algebraic approach. D. Reidel, HollandzbMATHGoogle Scholar
  7. Birkhoff G, Von Neumann J (1936) The logic of quantum mechanics. Ann Math 37:823–834MathSciNetCrossRefGoogle Scholar
  8. Butnariu D, Klement P (1993) Triangular norm-based measures and games with fuzzy coalitions. Kluwer, DordrechtCrossRefGoogle Scholar
  9. Bonzio S, Chajda I (2017) A note on orthomodular lattices. Int J Theor Phys 56:3740–3743MathSciNetCrossRefGoogle Scholar
  10. Dumitrescu D (1993) Fuzzy measures and the entropy of fuzzy partitions. J Math Anal Appl 176:359–373MathSciNetCrossRefGoogle Scholar
  11. Dvurečenskij A (2017) On orders of observables on effect algebras. Int J Theor Phys 56:4112–4125MathSciNetCrossRefGoogle Scholar
  12. Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer Academic Publishers, DordrechtCrossRefGoogle Scholar
  13. Kagan E, Ben-gal I (2011) Navigation of quantum-controlled mobile robots. In: Topalov A (ed) Recent advances in mobile robotics. InTech, Rijeka, pp 311–326Google Scholar
  14. Kalmbach G (1983) Orthomodular lattices. Academic Press, LondonzbMATHGoogle Scholar
  15. Keimel K, Lawson JD (2005) Measure extension theorems for \(T_0\)-spaces. Top Appl 149:57–83CrossRefGoogle Scholar
  16. Khare M, Gupta S (2010) Non-additive measures, envelops and extensions of quasi-measures. Sarajevo J Math 6(18):35–49MathSciNetzbMATHGoogle Scholar
  17. Khare M, Gupta S (2008) Extension of non-additive measures on locally complete \(\sigma \)-continuous lattices. Novi Sad J Math 38(2):15–23MathSciNetzbMATHGoogle Scholar
  18. Khare M, Roy S (2008a) Conditional entropy and the Rokhlin metric on an orthomodular lattice with Bayesian state. Int J Theor Phys 47(5):1386–1396CrossRefGoogle Scholar
  19. Khare M, Roy S (2008b) Entropy of quantum dynamical systems and sufficient families in orthomodular lattices with Bayesian state. Commun Theor Phys 50:551–556MathSciNetCrossRefGoogle Scholar
  20. Khare M, Singh AK (2008) Weakly tight functions, their Jordan type decomposition and total variation in effect algebras. J Math Anal Appl 344(1):535–545MathSciNetCrossRefGoogle Scholar
  21. Khare M, Singh B, Shukla A (2018) Approximation in quantum measure spaces. Math Slovaca 68(3):491–500MathSciNetCrossRefGoogle Scholar
  22. Kelley JL, Nayak MK, Srinivasan TP (1972) Premeasure on lattices of sets II. In: Symposium on vector measures, Salt Lake City, UtahGoogle Scholar
  23. Marczewski E (1953) On compact measures. Fund Math 40:113–124MathSciNetCrossRefGoogle Scholar
  24. Markechová D (1993) The entropy of complete fuzzy partitions. Math Slovaca 43:1–10MathSciNetzbMATHGoogle Scholar
  25. Ming LY, Kang LM (1997) Fuzzy topology. World Scientific Publ. Co., SingaporezbMATHGoogle Scholar
  26. Morales P (1981) Extension of a tight set function with values in a uniform semigroup. In: Kölzow D, Maharam-Stone D (eds) Measure theory, Oberwolfach 1981, Lecture Notes in Mathmatics, vol 45. Springer, pp 282–292Google Scholar
  27. Nayak MK, Srinivasan TP (1975) Scalar and vector valued premeasures. Proc Am Math Soc 48(2):391–396MathSciNetCrossRefGoogle Scholar
  28. Pap E (1995) Null-additive set functions. Kluwer Academic Publishers, DordrechtzbMATHGoogle Scholar
  29. Pták P, Pulmannová S (1991) Orthomodular structures as quantum logics. Kluwer, DordrechtzbMATHGoogle Scholar
  30. Riečan B (1979) The measure extension theorem for subadditive probability measures in orthmodular \(\sigma \)-continuous lattices. Commun Math Univ Carol 202:309–315zbMATHGoogle Scholar
  31. Topsøe F (1970a) Compactness in spaces of measures. Stud Math 36:195–212MathSciNetCrossRefGoogle Scholar
  32. Topsøe F (1970b) Topology and measure. In: Dold A, Eckmann B (eds) Lecture notes in mathmatics, vol 133. SpringerGoogle Scholar
  33. Varadarajan VS (1968) Geometry of quantum theory, vol 1. Van Nostrand, PrincetonCrossRefGoogle Scholar
  34. Yali W, Yichuan Y (2017) Notes on quantum logics and involutive bounded posets. Soft Comput 21:2513–2519CrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AllahabadAllahabadIndia

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